Vanilla 1.1.9 is a product of Lussumo. More Information: Documentation, Community Support.
This question strikes me as too subjective and argumentative for MO. I think a more acceptable version could be written (e.g. a reference request asking for essays that mathematicians have written about rigor), but that it is not the current version.
Of course, the canonical reference would be the series of essays inaugurated by Jaffe and Quinn in the BAMS. But something about the question saddens me: having to explain the fundamental value of proofs in mathematics is almost like having to explain the difference between right and wrong. To think of proofs as mere ritual is sad and degrading, and very limited in outlook.
I'm not seeing how the question could be re-written to be anything more than a wide-ranging discussion. I suspect Terry Tao's comment is the closest to a good response to that (not yet written) discussion topic.
Could someone propose a concrete rewriting of the question?
I admit I am not very familiar with the manner in which mathoverflow is run, and what exactly is expected of the community or of each question on the site. Nevertheless, I have seen several somewhat "subjective and argumentative" questions which gave rise to wide-ranging discussions. For a very recent example, see http://mathoverflow.net/questions/128130/what-does-a-mathematician-expect-from-mathematics-education (which is nevertheless less controversial for mathoverflow, I would probably say).
So let me phrase a more precise question. Can someone carefully explain to me what precisely makes the closed question under discussion innapropriate for mathoverflow? What is the motivation for closing the question? I admit I am very confused when people call this question (or most other questions, for that matter) "subjective and argumentative" or "not a real question", given that these criterions seem to be applied/enforced somewhat whimsically or unpredictably. I mean that this is a question which a few people find interesting or somewhat pertinent, judging by some of the ongoing discussion, even if controversial or perhaps tendentious. I am afraid there just aren't that many places on the internet --- actually, none that I know of --- where one can meaningfully discuss these questions with the appropriate group of people, i.e. mathematicians.
Can someone please enlighten me?
Have you read the FAQ and the "How to ask" documents, Ricardo? It might be helpful to put your discussion in the context of those guiding documents, rather than via anecdotal examples. There is a certain aspect of randomness to community moderation, so we should try to avoid the relativist drift of anecdotes.
@Ryan: Thanks for the helpful reference. I will read more carefully the "How to ask" page. In the meantime, it seems to me that that page only addresses (allows?) research questions. So let me try to ask a more precise question, if it does not lead this discussion too astray. How much latitude do the moderators give to "soft" questions which are not about mathematical research, yet have to do with mathematical practice, as happens with the question under discussion and a few occasional others? I apologize for not having any specific references to point to at the moment. If it helps, I can try to find some.
I am simply trying to determine more precisely how the rules in the "How to ask" page are enforced in practice. Thank you for your attention.
@Ricardo Andrade: some general and some specific remarks.
You asked how much lattitude the moderators give. I am not sure if this is just a slightly imprecise wording or if there is some misunderstanding: the moderators (in a strict sense; that is the six mentioned in the FAQs) are not really involved in most closing decisions, or at least they are not involved as moderators, but only as 'normal users'. The closing (and reopening) is handled on a day to day basis by at the moment about 300 users that have 3000+ points, so you are to be one of them soon.
Completely practically speaking a question is closed if 5 out of these 300 decide it should be (and reopened if 5 decide this way). This accounts already for a lot of fluctuation. This is not quite as strong as I put it as many out of the 300 hardly ever vote and some are even already inactive, but still there is a considerable pool of frequent voters that do vote.
Regading 'soft' questions in general: this is particullarly difficult to say what is the criterion here. I think I can claim that there are likely only very few people that followed this more closely than me over the last two years or so. I think I have meanwhile some general feeling what will work and what will not work, but still there are many surprises. I do not consider this as a big problem, since as you remarked, what is written in the official description concerns only mathematical question. I thus take the opinion everything else is a priori off-topic and sometimes an expception is made. And essentially by definition there cannot be preceises rules governing exceptions.
Regarding 'subjective and argumentative': what I consider as typically problematic is when answers (necessarily due to the question) will (essentially) purely be based on personal opinion as opposed to at least also containing a factual component. This is always problematic, but even more if the subject is potentially controversial.
Regarding 'not a real question': this can be used if the question is not really a question, as the reason says. But rather some attempt to start a discussion couched into the form of a question.
For me this is often a reason for closure: it seems OP does not really want to ask a question but rather means to start a discussion via some almost or sometimes truly rhetorical question. Some questions to me really read like: This and that. Surely, you agree?! That and this. Clearly this is scandalous. Isn't it?!
And there are still many more things. If you really want to get a feel, you could read old meta-discussion in the category 'Is this question acceptable?' You will find pages over pages of discussion. There is no rule not even a clear concensus, but mainly just a group of users each having their opinion on the matter. And then "we fight it out" (mainly in a friendly way though :-) ) on a case by case bases.
For the present question: let us just look at what is actually asked. Some sometimes find this a stupid way to proceed as anyway it is clear what is meant, but I do not follow this, since if it were so clear why not say it clearly.
There is the title question:
Is rigour just a ritual that most mathematicians wish to get rid of if they could?
As OP said this seems like a non-question, but now he had some experience that makes them doubt that (but nobody shared this specific experience, this is a problem! see below). So, then:
Has something happened in the world of mathematics that I am not aware of?
Hard to tell. For one thing, how should I know what OP is aware of. And also the preceeding description of the event is quite unclear to me. There are many many details missing to allow me to form a mental image of the situation to say anything in a meaningful way.
Do mathematicians not preach what they practice (or ought to practice)?
I do not even quite understand what this means. At least it seems based on OP's (personal and not communicated) idea what mathematicians ought to practise.
So, to me clearly there is no specific question to be answered here. The best one could do is start some general discussion about the importance or not of prrofs in mathematics. And, discussion is something that in general is something that is tried to be avoided on MO. Sometimes it still happens for one reason or another but there are various users that agree that this should be minimized and vote like so.
Final note: my paraphrase of the current question could be read as quite dismissive. This is not the intent. But I considered it as useful to exagerate a bit to better get across what are problems one can see in this question.
@quid: Thank you so much for your very useful comments. In fact, you did dispel my misunderstanding regarding how the closure procedure occurs: I was not really aware of who had the privilege or capacity to vote to close. Thank you for taking the time to explain these basic things to me. It was also quite helpful that you fleshed out your personal analysis of the question presently under discussion, and explained how it does not fit well with the usual, intended style of communication on mathoverflow (i.e. minimal discussion). I will certainly give due thought to your remarks, and may take your suggestion to browse through some related old meta threads.
This question reminds me of an MO-inappropriate question that's been bubbling around in my brain for a long time. It is, roughly:
"What does 'rigor' actually mean to adult mathematicians?"
[I don't mean "adult mathematician" to mean anything profound here and I certainly don't mean use it in an elitist way. I am just describing something that, sociologically speaking, seems to happen to students of mathematics and seems to stop happening at a certain point in someone's mathematical career.]
The idea behind the question is this: we throw the word "rigor" around a lot, but mostly in conversations with people who are not full-fledged mathematicians. Especially, the last time someone told me that my argument was "not rigorous", I was an undergraduate and the person was grading my homework. And I think that he really meant, "Sorry, but your argument is incomplete in a significant way. To complete it would require not just more lines of text but actually more work / an additional idea beyond what you have done."
I don't think that the reason that no one has said "not rigorous" to me in my adult mathematical life is that I've become so damned rigorous. I think it's because if I have a gap in my reasoning they so more directly and specifically. (On the other hand it is true that I still get referee reports calling passages in my writing "unclear", despite the fact that, you know, I'm such a clear writer. Often I want to reply in frustration "Just saying 'this is unclear' is not very clear! Please tell me more specifically what the problem is.") Nor do I use the word "rigorous" myself, either with adult mathematicians or with students or other outsiders. As above, it feels a little lazy to me: surely I can explain better what the problem is.
Nevertheless sometimes I hear people, including adult mathematicians, using "rigor" when they talk about mathematics. Often one hears rigor described as a continuum, alongside similarly ephemeral but generally believed to exist concepts like depth, elegance, power and so forth. Increasingly I suspect that I don't believe in rigor on a sliding scale. If you show me a proof in an area in which I am sufficiently qualified to follow it, I will either see that it is correct, see that you've made a mistake, see that you have a gap in your reasoning that is not routine to fill, or find your writing so unclear that I just can't follow your argument. Which of these means you're not being rigorous?
I did, by the way, read Jaffe-Quinn and the responses to it. I think they're about something different (which is much more relevant to the question being discussed in the thread than the present rant), namely they want to introduce a new kind of proof, so they rename standard mathematical proof "rigorous proof". This seems clearly in line with what I said above: conventional mathematics -- what do they call it in their article? Charles Mathematics?? -- is rigorous by its nature.
Pete: If you haven't already read it you may be interested in this brief essay from Terry Tao's blog. In particular, I like how characterizes the way most "adult" (I usually use the word "mature") mathematicians work as "post-rigorous", as opposed to "non-rigorous" or something similar.
Nor do I use the word "rigorous" myself, either with adult mathematicians or with students or other outsiders. As above, it feels a little lazy to me: surely I can explain better what the problem is.
Yes, it's certainly important to explain the problem in more detail than just saying an argument isn't rigorous. However, I think rigor is an important explanatory concept for distinguishing between arguments that could be refined to a formal proof and arguments that could not without a lot of extra work. For example, heuristic or simplifying assumptions, models that are literally incorrect yet have explanatory power (e.g., random models for the distribution of primes), uncontrolled approximations, techniques that work generically but fail in critical cases, failure to consider non-obvious cases as possibilities in the first place, etc.
There are students, amateurs, and people in other fields who genuinely do not understand this distinction, and it's valuable to be able to explain that there are all sorts of arguments, which may be illuminating or convincing without being fully rigorous, while only rigorous arguments count as a mathematical proof.
Dear Henry,
Thanks for your reply.
However, I think rigor is an important explanatory concept for distinguishing between arguments that could be refined to a formal proof and arguments that could not without a lot of extra work.
Okay, now I have a key question: which of these does "not rigorous" refer to? [I'm guessing you mean the latter, but I'm honestly not 100% sure.] Because I think I have heard it used in both ways. Even with regard to graders of my undergraduate problem sets, I only now think that they probably meant the latter. As I recall, when I would get "not rigorous" comments I would get partial credit for the answer, which at the time seemed more indicative of sloppiness than truly missing ideas. I also suspect that the ambiguity built into the term may be part of why people use it: i.e., it's not as harsh to say "this is not rigorous" as "you haven't proved anything yet" because the former could mean the latter but maybe it just means you skipped over some details.
And it is a strange term which is meaningful primarily in its absence. The Jaffe-Quinn idea of replacing the term "mathematics" with "rigorous mathematics" went over like lead balloon in part, I think, because of its linguistic awkwardness: the most standard state of affairs should get the unadjectival term. We should use positive terminology instead, e.g. speaking in terms of probabilistic models, heuristics, and so forth. (Although, good lord, mathematicians don't necessarily know what "heuristic" is supposed to mean either. I have a whopper of an in-person story that confirms that.)
Added: Looking back at the question this thread is supposed to be about, my present inquiry seems less irrelevant than I previously thought. "What do you mean by 'rigor' anyway?" has got to come into a discussion about whether it is dispensable, although I admit the possibility that most other people have a clearer bead on the term than I do.
Finally (for now), I just reread the question and: "Death to Euclid!" is a very funny thing to say.
Hmm, that's a good question, and I was overlooking this ambiguity. I meant the latter (i.e., unrigorous = not formalizable), but I agree that in undergraduate education calling an argument unrigorous often means "I know perfectly well how to fill in the details, but part of the exercise here was for you to demonstrate that you could fill them in, and you haven't done that convincingly enough."
"Death to Euclid!" is a very funny thing to say.
A web search reveals that Basil of Caesarea, in his Address to Young Men on the Right Use of Greek Literature, refers to someone vowing death to Euclid, but it's actually Euclid of Megara rather than Euclid of Alexandria.
I just want to point out that this is a rather old topic, so there would be a lot to say about this if a question were asked in the right way.
For example, Brouwer, the father of intuitionism, had a very interesting sense of rigor:
The point of view that there are no non-experienced truths [...] has found acceptance with regard to mathematics much later than with regard to practical life and to science. Mathematics rigorously treated from this point of view, including deducing theorems exclusively by means of introspective construction, is called intuitionistic mathematics.
On the other hand, Brouwer rejected formalism:
[T]he intuitionist can never feel assured of the exactness of a mathematical theory by such guarantees as the proof of its being non-contradictory, the possibility of deﬁning its concepts by a ﬁnite number of words [...] or the practical certainty that it will never lead to a misunderstanding in human relations.
Brouwer's introspective constructions are a very interesting things to study. A lot of Brouwer's writings center on the role and impact of the creative mind on mathematics and it is a very important part of what he calls rigor.
I think perhaps the most important aspect of rigor, beyond its precise meaning is it's an act where we demand that we translate our perhaps rather nebulous and personal ideas into something that is for all purposes a completely artificial and foreign language. Its the act of translation into this neutral language that allows us to expose flaws in the core idea. It's similar to the act of converting pseudo-code into code and then hitting the compile button.
[Retracted.]
@HJRW: may I ask you what in your opinion is the significant difference (regarding the points raised in your comment) between ex0du5's and Pete L. Clark's (and several of the follow-ups it created) contributions to this thread?
quid - fair point. That'll teach me not to read the whole thread! I retract my previous objection.
(Added: on reflection, I do think there are some differences. Pete's comment is phrased as a question, whereas ex0du5's are polemical. But I agree that there are several comments here that may be 'off topic', and I don't particularly want to get into an argument about it.)
@HJRW: thank you for the reply. I do not wish to start a discussion either; let me only say that I do not read ex0du5's contribution as that polemical, but by intent as constructive.
ex0du5, I was unable to find any specific indication of harm produced by insistence on rigor in your lengthy series of posts, despite several references to its existence. You also haven't really defined what you mean when by "progress" when you say that rigor acts as a constraint on it. Here is my attempt to clarify:
Suppose Mathematician X's claimed proof of some conjecture is viewed by the community as a brilliant reduction to a collection of technical conjectures that X couldn't be bothered to work out or write up completely. If we don't accept it as a complete proof, is that a lack of progress? Or, are you arguing that the reward and reputation system in mathematical society is structured in a way that people are often loath to use tools of questionable foundation (e.g., prime heuristics, Feynman path integrals, pre-Robinson infinitesimals) to make bold and illuminating conjectures?
It seems that you are opposing an unrealistic extreme position, namely valuing "complete rigor above all else" and foundations of absolute truth. Furthermore, the position you oppose seems to include the view that proofs that are conditional on a conjecture should be seen as valueless. I don't know anyone who takes such a position, and I don't think that is the subject of the question under consideration. The question asks whether proofs should be necessary for a mathematical result to be accepted as valid. It sounds more like a survey question than a MathOverflow question, so I'd rather not see it reopened.
Regarding rigor and harm, I think the stories of pilot-wave theory and infinitesimals say more about the harm done by academic communities ignoring non-trendy subjects and methods, than they do about rigor itself. Plenty of rigorous proofs are ignored because they are in unfashionable areas, even if they are later found to be important, and it is quite reasonable to view that missing time as harmful. In the examples, I don't see the harm coming from rigor:
1) The scenario I constructed, in which Mathematician X claims to prove a theorem, but the proof rests on unsettled conjectures that are swept under the rug, is a situation where non-rigorous reasoning would lead one to think that the theorem is settled, while sufficient rigor would lead one to correctly conclude that there was still work that needed to be done. Where do you see the harm from rigor? Perhaps if Mathematician X's arguments were rejected wholesale, and their value went unrecognized, that would be harmful, but I don't see why that is a necessary consequence of insisting on rigor. Indeed, what we get from a lack of insistence is situations where no one understands the details behind various theorems, and we thereby lose the ability to apply and generalize the methods with confidence.
2) de Broglie's 1927 pilot-wave theory appeared to be contradicted by the conclusion of von Neumann's 1932 theorem (note the reversal of your proposed chronology), but instead of blaming rigor for the lack of attention paid to the theory after that, I would place the blame more on the insufficiently rigorous examination of locality hypotheses, and perhaps even more on the politics of the physics community (which is not known for its insistence on rigor).
3) A proof using infinitesimals before Robinson is basically a proof of a theorem that is conditional on a conjecture, before the conjecture is proved. In such a situation, I don't see the harm in insisting that the proof of the theorem is unfinished, and that the theorem is not quite settled. Certainly, your story of correct statements proved using infinitesimals being derided as wrong describes harm, but in a more rigorous community, such claims of wrongness would need justification.
At any rate, I think the problems in the historical examples are in part due to a community's attachment to the appearance of rigor, rather than rigorous reasoning itself. I would welcome other examples of harm, but I am suspicious of the sort of patterns that we see in Joël's answer, where the romantic genius can't be bothered to justify all of the nontrivial steps.
There has only been one delete vote; they don't get reset when a question gets reopened. This question will not be deleted anytime soon.
@avocat: Accepting an answer is irrelevant regarding the form of deletion you mention.
Open again. But it seems to me that the raison d'etre of the question has been undermined, being to a large degree based on a misunderstanding of Milnor's point of view (as indicated by his quoted correction). There seems to be essentially zero support in the answers, and in Milnor's response, for the notion that mathematicians would gladly dispense with rigor if they could, the importance of rigor being acknowledged all around. Thus it feels to me that the answers, as elaborated so far, are mostly "preaching to the choir" (Zeilberger's speculations notwithstanding, which were offered merely as one data point, without any particular endorsement in the answer).
Interesting topic, perhaps, but IMO not a good fit for MathOverflow.
@ashgari.amir: on a practical note, could you please try to use less metaphoric and more direct formulations (or at least provide the latter in addition). I find it somewhat non-trivial to figure out what precisely you mean with your last comment, for example.
I'm also having trouble: believer in what to a skeptic about what? (I suspect that part of the problem is that the terms of the discussion -- what precisely is meant by "rigor" in the OP? -- have never been made clear, as noted earlier in this meta thread. That's one reason I don't think the question was too good for MO. That, and the confusing account of an event whose accuracy was contested by Milnor -- it just gives me a bad feeling.)
Let's call this one a draw.
I cleared all the remaining votes. The question should remain open until the time where it starts attracting junk answers, where it will be closed for that reason.
With 4 votes to close, I bet it would have been closed soon, which would have seemed a much better status: it's a hypothetical question based on a misinterpretation of what Milnor said, the phrasing is the epitome of "subjective and argumentative" (philosophical breakdown, death to Euclid, rigor as a ritual most mathematicians would get rid of if they could, etc.), and it is highly discussion-oriented. It's rare to see a question that could arguably be closed for every available reason except "exact duplicate", "blatantly offensive", and "spam". What puzzles me is the repeated votes to re-open - is there some reason I'm missing why this question could make more sense for MO than most closed questions? I can understand both sides of the arguments over soft questions in general, but I feel like there's some enthusiasm for this specific question beyond the support soft questions get in general, and I don't see where it is coming from. Can anyone explain?
I am in full agreement with Henry, and my hand had in fact wavered over the close button before François called it a draw. I wished I had had greater courage in my convictions.
P.S. Just to be clear, I'm not arguing for changing how things got resolved (it's not a big deal, especially with the question dormant), but I'm curious to understand other people's perspectives on the question and how they think MO should work.
@Henry Cohn: Let me try an explanation of what I think is present in favor of this question (while I am certainly not in favor of it; the only reason I did not vote to close it is that I do not vote to close anything at all at the moment).
The closest to an explicit argument in favor of the question is I think this relatively early comment (with 15 upvotes):
On my opinion, this is a legitimate and important question. These discussions are common, and sometimes even happen on the pages of BAMS. I propose to reopen. – Alexandre Eremenko
The first question to me is a bit what "this" [question] is but I would assume in the end the general subject of the importance and/or status of rigor and proof in mathematics is meant. Since this is actually a relevant and interesting topic one can be of the opinion that some answering/discussion/talk on it should happen on MO. What precisely is asked for or even meant by rigor is not so important; everybody could interpret it a bit as they see it personally and say something (interesting); the range of the topics in the answers given also supports this.
To me this is vaguely reminiscent of the situation in some earlier questions like http://mathoverflow.net/questions/101420/music-mathematical-point-of-view-revised-closed where in particular early versions of the question did not make much sense as asked and/or the given answers had little to do with the question as asked but some found it a good occassion to talk a bit about math and music in general.
In my observation/opinion this is not rarely a key-point in disagreement: the ones say/think the question is not good at least not as asked, the others say/think even if the question is not so good as asked one can/could interpret it in an interesting way so that the answers will/could be interesting.
Or even more roughly, the divide between: "let us answer questions" and "let us talk about subjects".
That there was so much voting on this one has I think also practical reasons not only intrinsic to the question; it was visible a long time (on main and on meta), and reappeared repeatedly. Yet it is also a general subject that many/most mathematicians care about in one form or another and so it is not surprising there was a big pool of people contributing their opinion (whereas perhaps the relation of math and music is of less universal interest in the community).
@bsteinberg: I didn't enter a vote to close lightly. But looking back at François's comment from seven days ago here in meta, it is my opinion that the question has begun to attract answers which, though I wouldn't term them "junk" exactly, reflect poorly on MathOverflow. I really strongly feel it is time that this question be closed. (Edit: I hadn't seen his stronger declaration "the war is now over" in the comments under the OP until just this moment.)
I am not sure Todd Trimble meant this (also), but to continue on this line of reasong: while in some sense I like fedja's answer, I cannot help but feel that sentences like:
[...]for the students it ultimately means that they are treated as subhuman beings, i.e., they are considered as having almost no intelligence whatsoever[...]
Many mathematicians lost all pride and turned into mere beggars for money [...] and recognition [...].
are not really appropriate for MO; and all the observations on miltatry, politics, and (modern) art in the mix.
Not quite clear where this discussion will go if it continues still a bit more. Already, the 'Entartete Mathematik' in a comment is something that in my opionion should really be avoided.
@quid: yes, I mainly meant fedja's answer. While I can sympathize with some of his sentiments, the discussion does not seem to be constructive (to put it very mildly). And some of the other recent answers seem either not very well thought through, or not all that relevant. (Oh, and I agree that "Entartete Mathematik" isn't helping matters at all.)
I am sorry to have to rehash this, but I think the problem begins with the question. Expanding on what Ryan Budney said way up top: the description of the panel discussion which forms the basis of the question is vague and anecdotal (and IMO, slightly alarmist), and it's basically impossible to know what happened at that event. As a result, the question became a Rorschach which invited an overly wide and meandering discussion.
Also: sorry to the moderators if I'm causing a pain in the neck by casting another vote to close after they wanted to stop the "war". But I think this merits reconsideration.